A uniqueness result for a semilinear elliptic problem: A computer-assisted proof |
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Authors: | PJ McKenna F Pacella |
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Institution: | a Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, MSB 328, USA b Dipartimento di Matematica, Università di Roma “La Sapienza”, P.le A. Moro 2, 00185 Roma, Italy c Institut für Analysis, Universität Karlsruhe, 76128 Karlsruhe, Germany |
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Abstract: | Starting with the famous article A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243], many papers have been devoted to the uniqueness question for positive solutions of −Δu=λu+up in Ω, u=0 on ∂Ω, where p>1 and λ ranges between 0 and the first Dirichlet eigenvalue λ1(Ω) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ=0. In this article, we prove uniqueness, for all λ∈0,λ1(Ω)), in the case Ω=2(0,1) and p=2. This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ1(Ω), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem. |
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Keywords: | 35J25 35J60 65N15 |
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